For a fantastic and unique experience you should try to eat at the fish boats in Eminonu.
Basically you have these beautiful typical Ottoman decorated boats, grilled fish and making delicious sandwiched with fresh salad and fresh grilled fish. You can seat down and ask for a drink or just take your sandwich with you.
This is a interesting way of eating out, but also a way to eat very cheap in Istanbul. I personally know people that eat 2 or 3 grilled sandwiches per day to spend less possible with food. The price for a sandwich is of 4 Turkish Liras.